Skip to main content
SpringerLink
Log in

On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider a general finitely ramified fractal set called a nested fractal which is determined byN number of similitudes. Basic properties of the integrated density of statesN (x) for the discrete Laplacian on the associated nested prefractal are investigated. In particulard N is shown to be purely discontinuous ifM<N, whereM is the number of branches of the inverse of the rational function involved in the spectral decimation method due to Rammal-Toulouse. Sierpinski gaskets and the modified Koch curve are special examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat.6, 103–144 (1965)

    Google Scholar 

  2. Fukushima, M.: Dirichlet forms, diffusion processes and spectral dimensions for nested fractals. In: Albeverio, S., Fenstad, J.E., Holden, H., Lindstrøm, T. (eds.), Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, Vol.1. Cambridge: Cambridge Univ. Press, 1992, pp. 151–161

    Google Scholar 

  3. Fukushima, M., Nakao, S., Kotani, S.: Random spectrum. Seminar on Probability, Vol.45, 1977 (in Japanese)

  4. Fukushima, M., Shima, T.: On a spectral analysis for the Sierpinski gasket. Potential Analysis1, 1–35 (1992)

    Article  Google Scholar 

  5. Kigami, J., Lapidus, M.L.: Weyl's problem for the spectral distributions of Laplacians on p.c.f self-similar fractals. Commun. Math. Phys.158, 93–125 (1993)

    Article  Google Scholar 

  6. Kusuoka, S.: Diffusion processes on nested fractals. Lecture Notes in Math. vol. 1567, Springer 1993

  7. Lindstrøm, T.: Brownian motion on nested fractals. Memoir AMS420, 1989

  8. Malozemov, L.: The difference Laplacian Δ on the modified Koch curve. Russ. J. Math. Phys.3, no. 1 (1992)

    Google Scholar 

  9. Malozemov, L.: The integrated density of states for the difference Laplacian on the modified Koch graph. Commun. Math. Phys.156, 387–397 (1993)

    Article  Google Scholar 

  10. Rammal, R.: Spectrum of harmonic excitations on fractals. J. Physique45, 191–204 (1984)

    Google Scholar 

  11. Rammal, R., Toulouse, G.: Random walks on fractal structures and percolation clusters. J. Physique Lett.43, L13-L22 (1982)

    Google Scholar 

  12. Shima, T.: On eigenvalue problems for the random walks on the Sierpinski pre-gaskets. Japan J. Indus. Appl. Math.8, 127–141 (1991)

    Google Scholar 

  13. Shima, T.: Lifschitz tails for random Schrödinger operators on nested fractals. Osaka J. Math.29, 749–770 (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Science, Faculty of Engineering Science, Osaka University, 560, Toyonaka, Osaka, Japan

    Masatoshi Fukushima & Tadashi Shima

Authors
  1. Masatoshi Fukushima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tadashi Shima
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H. Araki

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fukushima, M., Shima, T. On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals. Commun.Math. Phys. 163, 461–471 (1994). https://doi.org/10.1007/BF02101458

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02101458

Keywords

Search

Navigation